give a, b and c are rational numbers, show the expressions are rational.
i.) a^2 + b^2 + c ^2
ii.) (5a^3 +6b^4)/(a^2+b^2)
iii.) a + a^2 + a^3 + ... +a^10
I had alot of trouble proving this, and it was not enough work.
please explain these.
Speaking generally there are a lot of different operations one can do to rational numbers without leaving the rational world.
The product of any finite number of rational numbers is rational:
let be rational numbers.
say are integers.
then are integers
therefore is rational.
Same kind of proofs can be used to show the sum of 2 rationals is rational, which extends inductively to the finite sum of rationals. The division of rationals remaining rational is a direct consequence of the multiplication of rationals remaining rational. So it is not hard to see that any finite composition of such operations on rationals results in a rational. Proving these general statements might be easier than specific examples which can get very messy.